Question

How do you find the solution set for $y = 6x - 5$ and $y = - x + 9?$

Answer 1

Hint: This problem deals with solving the system of linear equations in $x$ and $y$. This is done by adding or subtracting a multiple of one equation to the other equation, in such a way that either the $x$-terms or the $y$-terms cancel out. Then solve for $x$(or $y$, whichever left) and substitute back to get the other coordinate.

Complete step-by-step solution:
Given a pair of linear equations in $x$ and $y$, which are given by $y = 6x - 5$ and $y = - x + 9$
Now we have to solve these two equations in order to get the values of $x$ and $y$.
Consider the given two equations and rearrange them to solve accordingly.
Consider the first equation, as given below:
$ \Rightarrow \;y = 6x - 5$
Rearranging the terms of the equation gives:
$ \Rightarrow \;6x - y = 5$
Now consider the second equation, as given below:
$ \Rightarrow x + y = 9$
Rearranging the above equation, so as to solve this equation together with the first equation, as:
$ \Rightarrow x + y = 9$
Adding the first and second equations as given below:
$ \Rightarrow \;6x - y = 5$
$ \Rightarrow x + y = 9$
$ \Rightarrow 7x = 5 + 9$
On simplification of the above two equations, as given below:
$ \Rightarrow 7x = 14$
$\therefore x = 2$
Substituting the value of $x = 2$, in the second equation, as given below:
$ \Rightarrow x + y = 9$
$ \Rightarrow 2 + y = 9$
Grouping the constants to the other side of the equation, gives:
$ \Rightarrow y = 9 - 2$
$\therefore y = 7$

The values of $x$ and $y$ are $2$ and $7$ respectively.

Note: Here a system of equations is when we have two or more linear equations working together. Here this problem can be done in another way but with a slight change with the method solved here. Here instead of adding both the equations, we can multiply the second equation with 6 and solve both the equations, finally ending up the same solution.